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Mathematics
Math at its core is about establishing truths separate from sensual qualities, seeking patterns based upon these truths, systematically removing contradictions/inconsistencies from the patterns, and formulating conjectures with all of the above in mind. It is the one true language apart from reality which ironically makes it so useful. Here we wish to provide resources which will help you to develop mathematical skills for where ever you choose to apply them. Precalculus Note that any autodidactic education requires a minimum amount of fundamentals, and to grasp the higher levels of math you absolutely need to understand the basic concepts known as precalculus, which is generally the math you will see up to high school. If you lack any of these fundamentals, you should refresh your knowledge at pages like Khan Academy or PatrickJMT. If you think you are fit, you can also directly start with calculus, although I would advise to skim Stewart's Precalculus before you do so. If you are a total beginner, it is recommended to do the interactive exercises on Khan Academy, because they are a really helpful tool to quickly refresh your school knowledge up until calculus. You should do all the chapters up to Precalculus, that is: Early Math, Arithmetic, Pre-Algebra, Basic Geometry, Algebra I, Geometry, Algebra II, Trigonometry, Probability and Statistics. You don't need to listen to every video, but you should cover each exercise once to check if you understand it. Once you finish the Precalculus module, you can continue with your first book. If you are still fit regarding math, you should at least do the Precalculus module on Khan Academy to be sure you have grasped everything necessary. For a general overview on the topics to come, choose any book on Precalculus, though I recommend the following: Stewart's Precalculus You can finish this book in a few weeks. It is already structured in a way that you can do 1-2 chapters per day for 6 days and do a review day on the 7th. You will be familiar with most concepts in this book, but especially if you just come out of High School or have just finished Khan Academy from zero, it will be a good exercise for you. Problem books These elementary problem books are meant for additional non-routine practice, challenges & puzzles, killing time, preparing for school competitions and exams, or to steal interesting questions from when you're teaching or tutoring students >:3 * Challenging Problems in Algebra by Posamentier * Challenging Problems in Geometry by Posamentier * The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics by Shklarsky, Chentzov, and Yaglom * 103 Trigonometry Problems: From the Training of the USA IMO Team by Andreescu and Feng * 104 Number Theory Problems: From the Training of the USA IMO Team by Andreescu * 105 Algebra Problems from the AwesomeMath Summer Program by Andreescu * 106 Geometry Problems from the AwesomeMath Summer Program by Andreescu * 107 Geometry Problems from the Awesomemath Year-Round Program by Andreescu * 108 Algebra Problems from the Awesomemath Year-Round Program by Andreescu * "Problems From the Book" and "Straight From the Book" by Andreescu * The Stanford Mathematics Problem Book by Polya and Kilpatrick * Sequences, Combinations, Limits by Gelfand, Gerver, Kirillov, Konstantinov, and Kushnirenko * Challenging Mathematical Problems With Elementary Solutions by A. M. Yaglom and I. M. Yaglom * Hungarian Problem Book I-IV containing the Eötvös Mathematical Competitions from 1894–1963 The following have more advanced problems at the university level up to preparing for qualifying exams during graduate school * The Green Book of Mathematical Problems by Hardy and Williams * The Red Book of Mathematical Problems by Williams and Hardy * William Lowell Putnam Mathematical Competition: Problems & Solutions: 1938-1964 * The William Lowell Putnam Mathematical Competition: Problems and Solutions 1965–1984 * The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary * Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Szekely * Problems in Mathematical Analysis I: Real Numbers, Sequences and Series by Kaczor and Nowak * Problems in Mathematical Analysis II: Continuity and Differentiation by Kaczor and Nowak * Problems in Mathematical Analysis III Integration by Kaczor and Nowak * A Collection of Problems on Complex Analysis by Volkovyskii, Lunts, and Aramanovich * Problems in Group Theory by Dixon * Berkeley Problems in Mathematics by Paulo Ney de Souza and Jorge-Nuno Silva * Problems and Solutions in Mathematics (Major American Universities PH.D. Qualifying Questions and Solutions) Some strategies on how to approach difficult problems to solve them exactly or heuristically: * How to Solve It: A New Aspect of Mathematical Method by Polya * How to Solve It: Modern Heuristics by Michalewicz and Fogel * Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin by Weinstein and Adam * Guesstimation 2.0: Solving Today's Problems on the Back of a Napkin by Weinstein and Edwards * Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan * Problem-Solving Through Problems by Larson * Putnam and Beyond by Gelca and Andreescu Calculus Some students struggle with calculus but honestly it is really straight forward compared to advanced subjects in math. As long as you pay attention and go in looking to learn, you should quickly end up joining the rest of the math students in calling it trivial in respect. Single Variable Calculus The standards texts (Stewart, Rogawski, et al.) you see required for college classes are in all honesty quite terrible since they are not written with self-study in mind but just as a collection of exercises and a review of the basic methods. Do ''use them to practice calculus by not as a means to learn it. For a well done intuitive approach using infinitesimals, which is the way everyone ends up thinking about calculus which is technically ''nonstandard ''but by no means mathematically incorrect, "Elementary Calculus: An Infinitesimal Approach" by Jerome Keisler is a fantastic and free public domain book. For a rigorous ''standard (δ-ε) approach to the subject, your options are "Calculus" by Spivak or the classic "Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra" by Apostol. To just learn the methods of calculus, there are plenty of lectures on calculus for you to choose from on Youtube. Multivariable and Vector Calculus Again, the usual suspects you'll find assigned in college courses make good exercise books but terrible introductions to the subject. You're options are the latter part of Keisler's book above or "Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability" by Apostol. The following texts take a slightly more rigorous approach and go deeper into the subject by covering differential forms and manifolds. Most single semester courses on vector calculus do not have time to reasonable cover this material and is usually skipped until later but this perspective greatly aids in the understanding of the subject. You can study this material either when you first learn multivariable calculus or when want second pass on the subject, after learning the basic methods, to improve you understanding while deepening your knowledge by generalizing what you've seen before. They can also be used as supplements to an advanced multivariable analysis course. * C. H. Edwards Jr.'s Advanced Calculus of Several Variables (Dover Book) * Hubbard and Hubbard's Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach * Harold M. Edwards' Advanced Calculus: A Differential Forms Approach Curves and Surfaces in ℝ² and ℝ³ This subject is the study of Geometry using the tools that you learned in Vector Calculus and serves as a preparation to more abstract approaches to Differential Geometry you'll see in the future. Most schools only quickly pass through the subject during multivariable calculus but it will help in the long run if you study the material early on. * Pressley's Elementary Differential Geometry * do Carmo's Differential Geometry of Curves and Surfaces Ordinary Differential Equations The standard text used in college courses is "Elementary Differential Equations" by Boyce and DiPrima, which many people do seem to like (not me however). A cheap and very good alternative is "Ordinary Differential Equations" by Tenenbaum & Pollard (published by Dover) which is perfect for self study. Another great book is "Differential Equations with Applications and Historical Notes" by Simmons. Linear Algebra Linear Algebra breaks up into 2 complementary sub-fields: Matrix Algebra/Computational Linear Algebra and Theoretical Linear Algebra/Finite Vector Space Theory. Oddly enough, you could study them in any order but canonically you're typically expected to learn some matrix algebra first and then transition to vector spaces and/or more applied topics second. The necessary prerequisite knowledge is just precalculus but some calculus knowledge is useful and may appear in a few examples. Matrix Algebra For a first exposure to the subject there really isn't that much to learn. You typically cover systems of equations, matrix operations, Gaussian elimination (also known as row reduction), LU decomposition, determinants, eigenvectors and eigenvalues, and diagonalization possibly with a few additional fluff subjects to round out a whole course. Many times you can pick up the material while studying calculus or ODEs (like with Apostol or Hubbard2's book) so you can just skip to more advanced material. Also, the introductory material in first few chapters of advanced textbooks are often good enough to learn matrix algebra from. But if you insist, a gentle introduction for learners with weaker math skills can be found in "Matrices and Linear Algebra" by Schneider and Barker. Those with slightly better math abilities can benefit more from "Matrices and Linear Transformations" by Cullen. A free book for students seeking a honors introduction to linear algebra (and basic proofs) text is "Linear Algebra Done Wrong" by Treil (Don't worry, the title is a pun on Axler's book title below). There's also a whole host of vulgarly over expensive textbooks used by college courses at this level but most of them aren't very good and even if they were, the 2 aforementioned books above are far cheaper thanks to them being published by Dover. Applied Linear Algebra For a first book in applied linear algebra, "Linear Algebra and Its Applications" by Strang is the standard text used but it tends to be one of those love it or hate it texts. If you fall into the hate it camp, then Meyer's "Matrix Analysis and Applied Linear Algebra" is a good alternative. After reading one of them, you'll be more than ready to move onto advanced Numerical Linear Algebra and Matrix Analysis textbooks. Finite Vector Spaces To get started on the theoretical side of linear algebra you obviously should be familiar with the basics of proofs. Once you are, theory side has a lot of classic and well loved textbook to choose from: * Linear Algebra by Shilov (Dover Book) * Finite Dimensional Vector Spaces by Halmos * Linear Algebra by Friedberg, Insel, and Spence * Linear Algebra by Hoffman and Kunze Of course there's also "Linear Algebra done Right" by Axler and on the one hand, the stuff he does is great... but on the other hand, he fucking HATES determinants and goes crazy avoiding them. Because of that you shouldn't use his book alone to learn from and you really should read Shilov alongside of it. But it certainly gives an unique development of the subject. Refreshers and Advanced Books Now if you want a challenge, start off with "Linear Algebra and Its Applications" by Lax. It is good for learning linear algebra for the first time if you're a hot shot freshman, using it as a second book on linear algebra, or as a 3rd refresher book for those who are entering graduate school. Another good 3rd book for deeper linear algebra study, and if you have the abstract algebra background for it, is Roman's "Advanced Linear Algebra". Partial Differential Equations Historically, the study of PDEs was a major impetus for the development of many results of analysis without which makes the study of PDE destined to be somewhat more trickier than what you've seen before in your studies. Be prepared to do some real work. For a quick primer on PDEs, "Partial Differential Equations for Scientists and Engineers" by Farlow is pretty good albeit somewhat shallow. Fuller undergraduate treatments can be found with: * Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems by Haberman (The best applied text at this level) * Partial Differential Equation: An Introduction by Strauss * Partial Differential Equations by Fritz John Now if you have the required background in analysis, you can really study the meat of PDEs in detail with the following: * Partial Differential Equations by Jost (Strong bias for elliptic equations) * Partial Differential Equations by Evans (The standard introduction text for graduate PDEs) * Introduction to Partial Differential Equations by Folland (An more intermediate graduate level PDEs book than Evans) * Partial Differential Equations I: Basic Theory; II: Qualitative Studies of Linear Equations; III: Nonlinear Equations by Taylor Proofs and Advanced Mathematics True mathematics involves proofs, lots and lots of proofs (cry more physicists). You really need to master the art of writing valid proofs that do not make careless unstated assumptions or unproven assertions because you think they are obvious, which in fact are often dead wrong or at the very least hold most of the meat of the proof in them, and following along when reading proof in mathematical texts which requires diligently filling in all the skipped steps and checking which assumptions could be removed/weakened or what fails when they are removed/weakened. At their core basic proofs are really easy, frequently just a matter of unwrapping the definition and following your nose, but getting into the right mindset for them might take the neophyte some practice in order to realize that fact. Therefore you should work through a book or two on proofs before moving onto advanced mathematics and then blaming those books for being bad because you lacked the prerequisite mathematical maturity from skipping this step. Some good books to learn proofs are: * A Transition to Advanced Mathematics by Smith, Eggen, and Andre * A Primer of Abstract Mathematics by Ash * Conjecture and Proof by Laczkovich (An excellent supplement to either of the above books that shows a larger variety of proofs in mathematics) * Proofs from THE BOOK by Aigner and Ziegler (Not a textbook on proofs but it is an excellent collection of well done and elegant proofs to appreciate and draw inspiration from) After this, books on set theory and mathematical logic make excellent next steps in getting better at proofs and abstract mathematics in general before moving on to the much more difficult subjects like algebra and analysis. Combinatorics, graph theory, linear algebra involving vector spaces, and number theory textbooks would then be the next level to practice on and are fairly easy to read at this stage of mathematical maturity. Since you will like be revising your proofs quite often, now would be an ideal time to finally learn LaTeX (pronounced "lay-tech") to typeset your proofs and future papers in. Books Math Textbook Recommendations Chicago undergraduate mathematics bibliography Amazon's "So you'd like to... Learn Advanced Mathematics on Your Own"